Multi-Objective Road Adaptive Control of anActive Suspension System
Guido Koch, Klaus J. Diepold and Boris Lohmann
Abstract In the design of automobile suspension systems, the classical conflictbetween minimizing vertical chassis acceleration to increase passenger comfort andkeeping the dynamic wheel load small in order to ensure safe driveability must befurther eased due to increasing customer demands. In order to moderate the conflict-ing suspension objectives, a switching controller structure for an active suspensionsystem is developed which schedules linear optimal regulators depending on thepresent dynamic wheel load and suspension deflection. The goal is to maximizeride comfort while the wheel load is below certain safety critical bounds and thesuspension deflection remains within given construction-conditioned limits. Stabil-ity of the switching control system is analyzed using a multiple Lyapunov functionapproach. The performance of the road adaptive suspension control system is com-pared with a linear controller and the passive suspension system in simulations topoint out the benefits of the developed control concept.
1 Introduction
An automotive suspension system is expected to provide an optimum of ride comfortfor the passengers as well as safe driveability of the car, i.e. guaranteed tire-road con-tact. While the comfort aspect can be characterized by minimum vertical chassis ac-celeration, the safety aspect requires a stiff, well damped coupling between vehicleand road in order to keep dynamic wheel load deviations small. Another objectiveis that the suspension deflection should always remain below the constructionallygiven limits in order to prevent impulse-like accelerations of the suspended massas well as excessive wear of the components. These three requirements are con-
Guido Koch, Klaus J. Diepold and Boris LohmannInstitute of Automatic Control, Technische Universität München, 85748 Garching bei München,Germany; E-mail: [email protected], [email protected], [email protected]
flicting, [2]. However, this conflict can be eased by controlled actuators in activesuspension systems.
Some works on suspension controller design present the idea of adapting thecontrol objectives and thus the controller dynamics to the current road excitation. In[1, 4, 5] switching controllers are presented that minimize either the vertical chassisacceleration or the suspension deflection. In [9], a control structure with adaptiveproperties has been presented using a wheel load adaptive skyhook-control conceptfor a semi-active suspension system.
The new approach in this paper is the design of a nonlinear controller structurethat adapts to the current road disturbance in order to optimize the suspension re-garding the three conflicting objectives comfort, safe driveability and suspensiondeflection limits. The main idea is that maximum comfort should be achieved whilesafety for every road excitation is preserved by keeping the dynamic wheel load andthe suspension deflection below specified critical bounds. This is accomplished by aswitching controller structure based on six linear quadratic optimal controllers anda switching logic. The required actuator force for control should be feasible.
The remainder of this paper is organized as follows: First, models for an act-ive and passive vehicle suspension are presented in Section 2 and performance re-quirements are specified. In Section 3, the controller structure and the calculationof the scheduling variables initializing the switching are presented. Stability of theswitched control system for the active suspension is analyzed in Section 4 using amultiple Lyapunov function approach. Finally, simulation results and a performancecomparison are presented.
2 Modelling and System Requirements
The lift movement of the suspension system can be modelled using the well-knownquarter car models illustrated in Figure 1 [7, 11]. In the high bandwidth active sus-pension considered here an ideal actuator is integrated.
The model of the passive suspension results from the active suspension modelif F(t) = 0. The state-vector x and the output vector y are introduced as x =
zw
zg
zc
cc
dc
mc
cw
mw
zw
zg
zc
F(t)
cw
dc
cc
mc
mw
Fig. 1 Quarter car models of the passive (left) and active suspension (right).
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Multi-Objective Road Adaptive Control of an Active Suspension System
Table 1 Notation and parameter values [7].
Model parameter Symbol Value Unit
Quarter car chassis mass (sprung mass) mc 256 [kg]Wheel assembly mass (unsprung mass) mw 31 [kg]Suspension spring stiffness cc 20200 [N/m]Tire stiffness cw 128000 [N/m]Sprung mass damping coefficient dc 1140 [Ns/m]Undamped uncoupled naturalfrequency of the sprung mass ωc =
√cc
mc8.88 [rad/s]
Uncoupled natural frequencyof the unsprung mass ωw =
√cwmw
64.26 [rad/s]
[ zc − zw, zc, zw − zg, zw ]T and y = [ zc, Fdyn, zc − zw ]T where Fdyn de-notes the dynamic wheel load force. With the control input u(t) = F(t) and dis-turbance input ud(t) = zg(t) the quarter-car model can be expressed as a statespace model in the form
x = Ax + bu + eud , y = Cx + du , (1)
A=
⎡⎢⎢⎣
0 1 0 −1− cc
mc− dc
mc0 dc
mc
0 0 0 1cc
mw
dc
mw− cw
mw− dc
mw
⎤⎥⎥⎦ , b=
⎡⎢⎢⎣
01
mc
0− 1
mw
⎤⎥⎥⎦ , e=
⎡⎢⎢⎣
00
−10
⎤⎥⎥⎦ , (2)
C=⎡⎣− cc
mc− dc
mc0 dc
mc
0 0 cw 01 0 0 0
⎤⎦ , d=
⎡⎣
1mc
00
⎤⎦ . (3)
The model parameters are given in Table 1.
2.1 Performance Requirements
In order to ensure maximum ride comfort, the rms-value of the vertical chassis ac-celeration ‖zc‖rms is to be minimized. The human sensitivity for vibration is fre-quency dependent and the most sensitive frequency range for mechanical excitationis 4–8 Hz [3]. Therefore, a fifth order shaping filter Gc(jω) with an amplitude char-acteristic as depicted in Figure 2 and a state space representation
xf = Af xf + bf zc , zc,f = cTf xf (4)
is introduced, [3]. Its impulse response is gc(t) such that zc,f = gc(t) ∗ zc.
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G. Koch et al.
0.1 0.5 1 3 8 30 100−20
−15
−10
−5
0
Frequency [Hz]|G
c(j
ω)|
[dB
]
Fig. 2 Amplitude characteristic of the shaping filter Gc(jω) [3].
An augmented plant model with state vector xreg = [ xT , xTf ]T is used to incor-
porate the shaping filter in the controller design (Section 3) such that the controlledvariables are yreg = [ zc,f , zc, Fdyn, zc − zw ]T . With h = [ 0 1 0 0 ] the aug-mented plant model is
Pxreg =[
A 0bf hA Af
]︸ ︷︷ ︸
Areg
xreg +[
bbf hb
]︸ ︷︷ ︸
breg
u +[
e0
]︸︷︷︸ereg
ud , (5)
yreg =[
0 cTf
C 0
]︸ ︷︷ ︸
Creg
xreg +[
0d
]︸︷︷︸dreg
u . (6)
Safety requirements
For stochastic road excitation the dynamic wheel load’s rms-value should bebounded as follows1
max(‖Fdyn‖rms
) ≤ �var = Fstat
3, (7)
where Fstat = g (mc + mw) denotes the static wheel load. To ensure safety forsingular excitation events like potholes, the primary control objective changes fromcomfort to safety, i.e. dynamic wheel load limitation, if
|Fdyn|Fstat
≥ �sing = 0.75 . (8)
1 This is derived from the 3σ -rule and assures, assuming a normally distributed stochastic disturb-ance signal, that Fdyn remains within ±Fstat for 99.7% of the time [10].
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Multi-Objective Road Adaptive Control of an Active Suspension System
Requirements on suspension deflection
A total of |�z| = 0.1 m maximum suspension deflection is used as a limit in thesimulations of this study. Hitting the limit is modelled by an increase in cc as follows
cc ={
cc for −0.1 ≤ zc − zw ≤ 0.115 · cc for |zc − zw| > 0.1 ∧ zc − zw > 0 .
(9)
3 Controller Design
For the road adaptive suspension control n = 6 linear quadratic optimal (LQR) con-trollers with different weighting matrices Qy for separate primary control objectivesare designed in order to analyze the potential of the concept. For the application ofthe classical LQR design formalism with output weighting in the cost functional
JLQR =∫ ∞
0
(yTregQyyreg + uRu
)dt (10)
with Qy = QTy ≥ 0 and R > 0 chosen according to Table 2, the cost functional
needs to be slightly modified due to the direct feedthrough term dreg present in theaugmented suspension model. The resulting cost functional resulting from Eq. (10)is
JLQR =∫ ∞
0
(xTregQxreg + 2xT
regsu + uRu)dt (11)
with Q = CTregQyCreg, s = CT
regQydreg and R = dTregQydreg + R. We substitute
the control input u = u + R−1sT x in Eq. (11) in order to remove the mixed term2xT
regsu such that the conventional LQR design formalism can be applied [6]. Theoptimal solution for each designed LQR-controller with weighting matrix Qy,i isstate feedback ui = −kT
i xreg with kTi = R−1(bT
regPr,i + sTi ) and Pr,i being the
symmetric, positive definite solution of the algebraic Riccati-equation
AT Pr,i + Pr,iA − (Pr,ibreg + si
)R−1(bT
regPr,i + sTi
) + Qi = 0 ,
Pr,i = PTr,i > 0 .
(12)
Wheel load adaptation
Figure 3 shows the control structure in which the LQR-controllers are implemen-ted. In the following the part of the switching logic is presented that schedules thecontrollers by the scheduling variable qadp(t) with 0 ≤ qadp(t) ≤ 1 depending onthe dynamic wheel load. The basic concept is described in [9] but is modified here
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G. Koch et al.
Table 2 Controller weights Qy,i , R = 1 for all controllers.
Qy,i Value Controller Type
Qy,1 diag(3.5 · 105, 0, 0.1, 0
)Most comfort oriented controller
Qy,2 diag(8 · 104, 0, 0.4, 0
)Comfort oriented controller
Qy,3 diag(4 · 104, 0, 0.7, 0
)Intermediate controller
Qy,4 diag(104, 0, 0.9, 0
)Safety oriented controller
Qy,5 diag(10, 0, 1, 0) Most safety oriented controllerQy,6 diag
(0, 5.2 · 107, 10−3, 3 · 1012
)Suspension deflection controller
-
kT1
kT2
kTn
y
xu
ud
σ (qadp, qsusp)
switching logic
Fig. 3 Switching state feedback controller structure.
in details. In case of “hard” switching between the five comfort/safety oriented con-trollers kT
i , i ∈ {1, . . . , 5} (see Table 2) the piecewise continuous switching functionis
σ(qadp (t)
) ={ �qadp(t) · 5 if 0 < qadp ≤ 1
1 if qadp = 0(13)
where �· denote Gaussian brackets also known as the ceiling function. The sixthcontroller is activated separately as described at the end of this section. The schedul-ing variable qadp(t) increases with increasing wheel load and correspondingly asqadp(t) ≈ 1, the most safety oriented controller kT
5 is chosen by the switching lo-gic.
The current value of the scheduling variable is determined by two adaptation-rates as qadp(t) = min(1, qs(t) + qf (t)). The slow adaptation rate qs(t) (illustratedin the lower branch of block diagram in Figure 4) is used to adapt the suspensioncontroller to different variances of the road excitation signal. The variance of thedynamic wheel load is
σ 2Fdyn
= limT →∞
1
T
∫ T
0F 2
dyndt.
After Laplace transformation we replace the integral term 1s
by a first order low passfilter thus approximating the dynamic wheel load’s variance by
σ 2Fdyn
(s) ≈ 1
τss + 1F 2
dyn(s).
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Multi-Objective Road Adaptive Control of an Active Suspension System
1
0-
1
0
1
0
fast adaptation
slow adaptation
h(u)
u2
x
1τf s+1
1τs s+1
�2var
F 2stat
es
ef
qs
qf
gs
qadp
FdynFstatFdyn
Fstat= cwx3
Fstat
Fig. 4 Wheel load adaptation structure.
For τs the chassis mass eigenmodes’ time constant is chosen resulting in τs =2π
√mc/cc ≈ 0.71 sec.
The dynamic wheel load filtered in this manner is compared to the upper bound�var given by Eq. (7). The deviation
es(t) =σ 2
Fdyn(t)
F 2stat
− �2var
F 2stat
is scaled by a constant gs = 1.5 and is integrated by an output-limited integratorwith an output signal range of [0, 1] resulting in qs .
If the vehicle hits an occasional pothole on an otherwise smooth road, it is im-portant that an instant switching to a safety oriented controller occurs. Therefore,it is necessary to introduce a fast adapting term qf (t) in the calculation of qadp(t)
which is shown in the upper branch of the block diagram (Figure 4).To ensure that this fast adaptation part remains inactive as long as the relative
dynamic wheel load has not reached �sing = 0.75 defined in Eq. (8), the wheel loadis scaled by a nonlinear funtion h (based on a fourth order polynomial) shown inFigure 5 which output value ef is only nonzero if |Fdyn|/Fstat > 0.75. Again a lowpass filter for ef with a time constant
τf = 5
9· 2π
√mw
cc + cw
≈ 0.05 sec
is used. The output qf is limited to a range of qf ∈ [0, 1] as well.
Suspension deflection adaptation
A second scheduling variable qsusp(t) is calculated which determines when thesixth controller that suppresses excessive suspension deflection is activated by
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−1 −0.5 0 0.5 10
0.2
0.4
0.6
0.8
1
ef
FdynFstat
−1 −0.5 0 0.5 10
0.2
0.4
0.6
0.8
1
esusp
zc−zw|�z|Fig. 5 Nonlinear scaling function h(Fdyn/Fstat ) for fast wheel load adaptation (left) andf (zc − zw/|�z|) for suspension deflection adaptation (right).
σ(qadp, qsusp). This is described in detail in Section 4 because its switching be-haviour determines the stability of the system. The scheduling variable qsusp(t) iscalculated similarly to qf (t) with zc − zw/|�z| as input. Again a nonlinear functionf (zc − zw/|�z|) shown in Figure 5 is used being nonzero if |zc − zw|/|�z| > 0.66.The filter’s time constant is τsusp = 1/25 τf to ensure quick activation of the sus-pension deflection controller.
4 Stability Analysis
Stability analysis of switched systems is a very important issue because it is notguaranteed that switching between asymptotically stable subsystems (here: result-ing from different controllers) results in an asymptotically stable switched system.A well known stability analysis approach uses quadratic Lyapunov functions of theform V (x) = xT Px, V (0) = 0, V (x) > 0, V (x) < 0, P = PT > 0. If a mat-rix P can be found such that these equations are fullfilled, the equilibrium x = 0of the switched system is uniformly asymptotically stable, [8]. For numerical sta-bility analysis, the Lyapunov function and the condition for P has been formulatedas a pair of linear matrix inequalities (LMI) for every closed loop system matrixAcl,reg,i = Areg − bregkT
i of the switched active suspension system
(AT
cl,reg,iP + PAcl,reg,i
)< 0 for i ∈ {1, . . . , n} (14)
P = PT > 0 . (15)
These equations are solved numerically. Although no feasible solution for all sixcontrollers exists, two feasible solutions P1 and P2 have been obtained for two sub-sets of LQR-controllers kT
i with i ∈ {1, . . . , 5} and kTj with j ∈ {3, . . . , 6}. Switch-
ing between controllers within each subset thus results in an asymptotically stablesystem for arbitrary switching signals σ(·) because the Lyapunov function is a com-mon quadratic Lyapunov function (CQLF) of all systems within the subset [12].
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Multi-Objective Road Adaptive Control of an Active Suspension System
Soft switching
Because the active suspension system is improper regarding the control input,a discontinuous control force resulting from a noncontinuous switching functionσ(qadp(t), qsusp(t)) would result in discontinuities of zc and thus decreasing pas-senger comfort. To avoid these discontinuities “soft” switching by interpolatingbetween the controllers via qadp(t) and qsusp(t) directly is preferable. Therefore,stability for linear interpolation between the state feedback controllers is analyzed.
Theorem 1. If two closed loop system matrices A1 and A2 resulting from controlloops with different state feedback controllers K1 and K2 for the same open loopsystem matrix Aol have a CQLF characterized by P : P = PT > 0, the systemmatrix
Acl = µA1 + (1 − µ)A2 , 0 ≤ µ ≤ 1 (16)
being a linear interpolation of A1 and A2 has the same CQLF characterized by P.
Proof. A CQLF of the two closed loop systems is defined by
∃P : ATi P + PAi < 0 , P = PT > 0 ∀i ∈ {1, 2} . (17)
If we add the scaled Lyapunov functions for the closed loop system matrices we get
µ (AT1 P + PA1)︸ ︷︷ ︸
<0
+(1 − µ) (AT2 P + PA2)︸ ︷︷ ︸
<0
< 0 . (18)
For state feedback u = −Kix, i ∈ {1, 2} the closed loop system matrices havethe form Ai = Aol − BKi . Considering this and the fact that linear interpolationbetween the controllers Ki would result in the state feedback gain matrix K =µK1 + (1 − µ) K2, Eq. (18) can be transformed into
[Aol − BK]T P + P[Aol − BK] < 0 . (19)
�
Multiple Lyapunov function approach for stable suspension deflection control
Because no CQLF could be obtained for the whole set of closed loop suspensionsystem matrices Acl,reg,i , the stability for the switching control using all six con-trollers can be ensured using a multiple Lyapunov function approach, [12]. Asymp-totic stability in the sense of Lyapunov while switching smoothly between the twonumerically calculated Lyapunov functions V1(x) and V2(x) is ensured by two con-ditions:
1. Switching between the two Lyapunov functions is only allowed if the LQR-controller being activated by the switching is either kT
3 , kT4 or kT
5 because bothLyapunov functions are valid for these controllers.
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2. It is only allowed to switch back to the Lyapunov function Vl(x) with l ∈ {1, 2}at time t2 if the associated Lyapunov function has decreased since leaving it attime t1 (with t1 < t2), i.e. Vl(x(t2)) < Vl(x(t1)) [12].
Due to the same quadratic structure of V1(x) and V2(x) for our control problem, itis sufficient if condition 2 is fulfilled for l = 1 to guarantee asymptotic stability.
Switching to V2(x) is only necessary if the suspension deflection controller kT6
should be activated. Ensuring that condition 1 is satisfied, the wheel load adaptationparameter qadp(t) is smoothly increased automatically between 0.66 ≤ qsusp(t) ≤0.77 (see Figure 5) such that qadp(t) is at the lower limit (qadp(t) = 0.5) of theactivation of controller kT
3 at least before the suspension controller kT6 is enabled to
be switched to. For 0.8 ≤ qsusp(t) ≤ 0.9 the suspension controller kT6 is definitely
activated (with a similar function as f ( zc−zw
|�z| ) in Figure 5). Switching back to V1(x)
is only possible if qsusp(t) < 0.77 and additionally condition 2 is fulfilled for l = 1.
5 Simulation Results
The performance of the designed controller is compared to that of a conventional,comfort focussed LQR-controller with Qy,LQR = diag
(2.5 · 104, 0, 0.4, 0
)and
the passive suspension. As excitation signal zg(t) a superposition of two syntheticsignals (bumps) and two subsequent real measured road track signals is used (Fig-ure 6). The road excitation is zero for all points in time not depicted. Figure 7 showsthat the power spectral density (PSD) of zc is significantly reduced in the range ofthe chassis’ resonance frequency (approx. 1.41 Hz). In the frequency range from 4–8 Hz the adaptive controller performs best concerning comfort (approx. 17% betterthan the passive suspension and 8% better than the LQR-controller). The comfortgain vs. the LQR-controller would be more significant if a fully active suspensionmodel without passive suspension would be considered. Only the adaptive controllerkeeps the suspension deflection limit at the first bump.
The proposed road adaptive controller achieves a comfort gain in an rms-senseof approx. 20% in comparison to the passive system and of 11% compared to theLQR-controller for the simulated road profile (Table 3). All control forces are in anacceptable range.
Table 3 Performance of the road adaptive controller.
Quantity Unit Passive LQR Road adaptive Comment
‖zc‖rmsms2 3.06 2.75 2.45
‖gc ∗ zc‖rmsms2 2.85 2.56 2.36
‖Fdyn‖rms N 960.29 855.26 871.36 Limit: �var = 938.49 N‖F(t)‖rms N - 156.19 262.66max(|F(t)|) N - 2781 1771
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Multi-Objective Road Adaptive Control of an Active Suspension System
0.8 1
−0.1
−0.05
0
20 40
−0.05
0
0.05
44.55 44.65 44.750
0.02
0.04
0.06
0.08
50 60
−0.02
0
0.02
t[sec]t[sec]
t[sec]t[sec]
zg
[m]
zg
[m]
zg
[m]
zg
[m]
0 10 20 30 40 50 600
0.2
0.4
0.6
0.8
1
qadp(t
)
t[sec]0.9 0.95 1
0
0.2
0.4
0.6
0.8
1
qsusp
(t)
t[sec]
Fig. 6 Road signal zg(t) (left) and adaptation parameters qadp(t) (middle), qsusp(t) (right).
0 20
1
2
3
4
5
6
0.88 0.90.08
0.085
0.09
0.095
0.1
t[sec]f [Hz]
PSD
z c
zc−z
w[m
]
0 1 2 30
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 105
9 9.5 10 10.50
0.5
1
1.5
2
2.5
3x 105
f [Hz]f [Hz]
PSD
Fdyn
PSD
Fdyn
Fig. 7 Power spectral density (PSD) of the chassis acceleration zc (left), suspension deflectionzc−zw at the first bump (middle left), power spectral density of the wheel load zc for two frequencyranges (right); Adaptive controller (solid), LQR-controller (small dots), passive setting (large dots).
6 Conclusion
A road adaptive suspension controller has been designed that switches smoothlybetween different LQR-controllers depending on the current dynamic wheel loadand suspension deflection. The primary control objective is to maximize passengercomfort while given limits for the dynamic wheel load and suspension deflection arenot violated. Stability of the system is guaranteed by a multiple Lyapunov functionapproach implemented in the scheduling algorithm. The result is a performance gainof approx. 20% vs. the passive system and 11% vs. the LQR-controlled suspension.Only the road adaptive controller does not exceed the suspension limit.
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